Constructions - making accurate mathematical drawings

Learning Objective:

• we are learning to make accurate mathematical drawings using a limited range of equipment and using the mathematical properties of the shapes to ensure they are accurate.

Investing:

• You can see constructing shapes as a mathematical game: in chess each piece has a limited range of moves and likewise in construction there are a limited number of pieces each of which has a range of moves
• Alternatively you can see constructing shapes as a mathematical art: if a sketch of a shape is like a cartoon of a face, drawing your attention to particular key features, then a construction is like a portrait - it aims to be an accurate representation of the subject, but also conveys an understanding that is in the heart or mind of the artist. In the same way you might ask "why am I not allowed to just use a ruler and measure this?" in construction, you could equally ask "why not take a photograph ?" to get a portrait. The answer is the same - to make a portrait in a particular medium or a construction with limited tools forces the creator to show a deeper understanding of the subject and conveys meaning as well as representation.
• These skills are usually tested on UK GCSE exams and is a ‍‍US national standard in grades 9-12‍‍.
• This skill leads to an understanding of congruence in triangles and other figures.
• A 'free gift' with this skill is that we will also develop an improved understanding of loci.
• This could help you if you want to work in art, graphic design, product design, engineering and archtiecture: although many professionals in these fields now use computer-based tools to create accurate drawings, an understanding of the underlying principles may help you produce better, more creative ideas.
• perpendicular bisectors are used to construct Voroni diagrams such as this one showing the nearest international airport to each point in the world

Preparing:

• Can you already:
  • measure a line accurately to the nearest millimetre using a ruler?
  • measure an angle accurately to the nearest degree using a protractor (angle measurer)?
  • draw a line accurately to the nearest millimetre?
  • draw an angle accruately to the nearest degree?
  • set up your compasses and draw smooth, consistent circles with a given radius accurate to 1mm?
• Let's be sure you can: try sections 2c to 4c in the assessment below

constructions AfL test.doc

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• Assess yourself using the worked solutions:

constructions AfL worked solutions.doc

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• Before you start it would help to know:
  • that all points on the circumference of a circle lie the same distance (the radius) from the centre.
  • the key vocabulary: circle, centre, arc, intersection, line segment, parallel, perpendicular, bisector
• You will have a deeper understanding if you also know:
  • a little about loci and the properties of shapes

Discovering:

• Can you figure it out yourself from these examples?

online construction tool from Suffolk maths

construction toolbox.mht

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• Try to predict the next one aloud or on a mini-whiteboard.
• Investigate...

Modeling:

• Here are some examples of people getting it right:
• Here are some examples of people getting it wrong in typical ways:
• Here are some more examples. Did they get it right or wrong? Explain how you know!

Discussing:

• What would this one be? Tell your learning partner. Convince them you're right.
• Explain how you know.
• How would you explain this to someone who was new to it?

Explaining:

• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by...

Practicing:

• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

Sharing:

• A web page or wiki I have created to explain this can be found at...
• A presentation I have created and rehearsed looks like...
• A poster I have drawn or model I have made can be found...

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Next we could learn...
• This leads to...
• Now try...